What means "anything"? To me, this is a very poor description of what this sensor is supposed to do. It says infrared and motion sensor, but it looks like a PIR [passive infrared]sensor, and if it is, then it only responds to "heated" objects moving - like humans and maybe animals and maybe flames.

For the price, you can buy one and see if responds to "anything" moving, or only to humans, etc.

Since the documents only say that a HIGH output is 4V and don't specify a current limit I would look for a relay that will activate on 4V at 20 mA.

> in comparison to the magnet I described, 0,81wire, 40m 1v 1Amp

Look for a relay whose contacts can handle 1A r more at 1V or more.

> is it all, the battery, the IR detector, and the relay, on 4v or adding a transistor if on 5v. no controller.

Correct. No controller needed if you only want to close a relay contact. The sensor will run on 5V. The output of the sensor is enough to activate a transistor. If you have 5V for the sensor you also have 5V for the relay. The transistor can be chosen to provide as much current as the relay needs so the relay coil current is not critical. That will allow you to pick a less expensive relay.

If it is actually PIR, the way they detect motion is by having 2 sensors and detectingthe difference in readings between the 2, as when someone is moving. I wouldassume hot air will be picked up equally by both sensors, and therefore the differencein readings will be zero, but a moving candle being small should activate it [I think,but I've never tried a candle].

When you get the sensor, you should do some tests, such as seeing if it will detecta moving object that is not hot, like a large piece of cardboard waved in front of it,or whether it only responds to warm body moving.